Although the topic of this page is nuclear structure it is conveniently
instructive to introduce the problems of nuclear structure by looking at the
model of atomic electron structure. Atomic electron structure is a theory
that is well developed and well understood and, in fact, is the basis for
looking for a shell structure for nuclei.

The Model of Atomic Electron Structure

If the explanation for atomic electron structure had not developed from
the perspective of explanaining spectroscopic regularities it might have
emerged from investigating such phenomena has ionization. The ionization
potential of an element is the amount of energy, usually measured in volts,
required to remove an electron from an atom of that element. For example,
the ionization potential for hydrogen is about 13.6 volts. Removing an
electron from an atom of helium, on the other hand, requires about 24.6 volts.
To remove a second electron from that atom of helium would require about
54.4 volts.

The data on ionization potential for all the elements are shown in the
graph below. These data are for removing only one electron in each case.
The source of the data is the Handbook of Physics and Chemistry. A
value is not given for element Astatine (At), atomic number 85, and an interpolated value
is used in order to not create a problem in the display of the data.

As is seen in the above graph, the ionization potential reaches a peak
for certain levels of the atomic number and then abruptly falls to a
minimal level. The interpretation is that the electrons form shells and
when a shell is filled the configuration is exceptionally stable and
requires a lot of energy to knock an electron out of a filled shell. On the
other hand, an electron in excess of a filled shell is very easy to remove
from the atom. The marked peaks in ionization potential are those of the
noble gases; helium, neon, argon, xenon and radon. The inertness of these
elements is a consequence of the stability of the filled shells. The elements
one electron beyond a filled shell are the alkalai metals; lithium, sodium, potassium,
cesium, rubidium and francium. Hydrogen may also be consider a member of this
group.

The atomic numbers of the noble gases are marked in the graph. There values
are 2, 10, 18, 36, 54 and 86. These can be considered magic numbers for
electron structure stability. The differences in these numbers are:
8, 8, 18, 18, 32. These differences are twice the value of the squares of integers;
i.e., 2(22), 2(22), 2(32), 2(32), 2(42).
The first number 2 in the series {2, 10, 18, 36, 54, 86} is also of the
form of twice the square of an integer, 2(12).

The explanation of the magic numbers for electron structures is that
there are shells for 2(n2) electrons where n=1, 2, 3, 4...
The reason for the coefficient 2 in the formula is that there are two spin
orientations of an electron.
Pauli's exclusion principle operates and so electrons fill the states
sequentially with no two electrons of an atom in the same state.

There is a slight problem with the above presentation. Element 80,
mercury, also represents a local peak in ionization potential. The explanation
for mercury would require too much attention to the problem of electron
structure for the purpose of this introduction. The purpose of this
introductory material is to show what the evidence for a shell structure
looks like; i.e., characteristics reaching one extreme for some atomic
numbers and then immediately shifting to the opposite extreme for the
next larger atomic numbers.

However it is notable that secondary peaks occur for atomic numbers 30 and
48 as well as 80. The numbers 30 and 48 are the sums of three previous peak numbers; i.e.,
30=18+10+2 and 48=36+10+2. The number 80 does not quite fit this scheme, but the crucial
quantities are the shell capacities rather than the magic numbers of electronic structure.
These numbers are merely the sum of the shell capacities, {2, 8, 18, 32, 50, 72}. Thus
the number 80 is 50+18+8+2+2. The numbers 30 and 48 are 18+8+2+2 and 36+8+2+2, respectively.
Thus there is pattern to the secondary peaks.

There are other points of relative maxima at {4, 12, 26, 46, 64, 70}. Four is of course
2+2. Twelve is 8+2+2. Twenty six is 18+8 and 46 is 36+8+2. Sixty four is 32+32 and
70 is 50+18+2. What this suggests is that when complete shells cannot be filled there
is an advantage to filling a lower degree shell rather than having a partially filled
higher degree shell.

Models of Nuclear Structure

It is generally believed that nuclei are composed of protons and neutrons, although
there is some question as to whether a nucleus contains protons and neutrons as separate
substructures. It could be that a nucleus is just a bag of the quarks that compose
the protons and neutrons. At one time some thought that a nucleus was composed of substructures of
alpha particles (helium nuclie). For the analysis here a nucleus is considered as being composed
of protons and neutrons.

Let Z represent the number of protons in a nucleus and N the number of neutrons. The atomic number A is the
sum of Z nd N, the total number of nucleons in a nucleus. Let &nbsp:ZmN denote the
mass of a nucleus composed of Z protons and N neutrons. The mass of a proton in this notation is
&nbsp:1m0 and that of a neutron &nbsp:0m1.

One might expect the mass of a (Z, N) nucleus to be Z 1m0+N 0m1,
but generally that is not the value of ZmN. The difference is called the mass deficit Δ

ZΔN = Z1m0 + N0m1 − ZmN

When the mass deficit is expressed in energy units it is called the binding energy of the nucleus. Usually
this binding energy is expressed in millions of electron volts (MeV).
The binding energy for a nucleus is the amount of energy that would be required to disintegrate it into its component nucleons.
It represents the amount of energy gained by putting the component nucleons into that nucleus' particular structure.

The study of nuclear shell structure began with an investigation of the
changes in the binding energy of nuclei as the neutron number is changed
by one.

Information on how strongly a particular nucleon is bound can be gained by looking at the generic reaction

The graph of the above data indicates a dependence on the odd-even-ness of the proton and neutron number.
This is not a dependence on the equality of the proton and neutron number which is equal for all the cases considered.

There are a variety of other models of nuclear structure. The most prominent is
the liquid-drop model. This model is embodied in the
Semiempirical Formula for Nuclear Masses,
which gives good explanation for nuclear masses except for isolated cases
such as the alpha particle, the 2He4 nucleus.
It appears that there is some sort of shell structure that accounts for
the special cases that are not properly predicted by the semiempirical
mass formula.

Magic Numbers

Maria Goeppert-Mayer and other physicists examining the properties of the isotopes of elements discerned that isotopes in which the proton
and/or the neutron numbers were particular values have notable properties such as stability. These magic numbers are

Magic Numbers

2

8

20

28

50

82

126

The magic numbers for atomic electron structure are perfectly explainable in terms of a formula. The increments
in the particular stable atomic numbers are equal to twice the square of an integer. These increments correspond
to the maximum occupancy levels of shells. Therefore in looking for an explanation of the nuclear magic
numbers it is reasonable to look at the increments in the magic numbers; i.e.,

Magic Numbers

2

8

20

28

50

82

126

Increments

2

6

12

8

22

32

44

The difference in consecutive magic numbers could be sums of shell occupancy levels. For example, the value
of 44 might correspond to two shells of 22 each. Likewise a value of 22 might correspond to two shells, one of
occupancy 20 and one of occupancy 2.

The maximum occupancy levels for the atomic electron shells being the square of an integer n corresponds to the
sum of the first n odd numbers. Let look at the sum of the first even numbers.

Even Numbers

2

4

6

8

10

12

14

Cumulative Sum

2

6

12

20

30

42

56

The cumulative sum sum values show up in the set of increments of nuclear magic numbers and the increments
can be represented as simple sum of these cumulative sum values, as is shown below.

Increments ofMagic Numbers

2

6

12

8

22

32

44

Generation fromCumulative Sumsof Even Numbers

2

6

12

2+6

2+20

2+30

2+42

If there is anything to this pattern the number 14=2+12 should be something like a magic number increment (rather than a magic number).
This would mean that 22 and 34 should be nearly magic numbers.
Titanium, atomic number 22, has five stable isotopes but scandium, the element with atomic number 21, has only one
and vanadium, the element with atomic number 23 has only two.
Selenium, atomic number 34, has six stable isotopes but bromine, the element with atomic number 35, has only two
and arsenic, the element with atomic number 33 has only one.
This is suggestive of the magic-ness of 22 and 34 and thus that 14=2+12 is a magic number increment.
The numbers 6 and 14 would represent the maximum occupancy of some shell structure. If higher
shell structures can appear without all the lower shell structures being filled then 6 and 14 could
themselves be considered to be magic numbers.

The Stable Isotope and Isomer Data

The number of stable nuclides for magic numbers compared to the adjacent numbers look impressive but, as the table below shows, there is an even-odd alternation in
the relationship between the number of stable isotopes and the atomic (proton) number.

ProtonNumber

1

2

3

4

5

6

7

8

9

10

Number ofStable Isotopes

2

2

2

1

2

2

2

3

1

3

ProtonNumber

11

12

13

14

15

16

17

18

19

20

Number ofStable Isotopes

1

3

1

3

1

4

2

3

2

5

ProtonNumber

21

22

23

24

25

26

27

28

29

30

Number ofStable Isotopes

1

5

1

3

1

4

1

5

2

5

ProtonNumber

31

32

33

34

35

36

37

38

39

40

Number ofStable Isotopes

2

4

1

5

2

5

1

4

1

4

ProtonNumber

41

42

43

44

45

46

47

48

49

50

Number ofStable Isotopes

1

6

0

7

1

6

2

6

1

10

ProtonNumber

51

52

53

54

55

56

57

58

59

60

Number ofStable Isotopes

2

4

1

9

1

6

1

2

1

5

ProtonNumber

61

62

63

64

65

66

67

68

69

70

Number ofStable Isotopes

0

4

1

6

1

7

1

6

1

7

ProtonNumber

71

72

73

74

75

76

77

78

79

80

Number ofStable Isotopes

1

5

2

4

1

5

2

5

1

6

ProtonNumber

81

82

83

84

85

86

87

88

89

90

Number ofStable Isotopes

2

3

The odd-even alternation indicates a pairing of protons within the nucleus; perhaps the existence of alpha
particle subsystems.

The average number of stable isotopes increases with proton number reaching a peak for proton number 50
(Tin) and declines generally thereafter, as shown in the table below.

ProtonNumberRange

0's

10's

20's

30's

40's

50's

60's

70's

80's

90's

AverageNumber ofStable Isotopes

1.7

2.3

2.8

3.0

3.4

3.7

3.2

3.3

1.1

0.0

The real test of the magic-ness of a number is whether it appears so in terms of the neutron number. The numbers of
stable isotopes as a function of neutron number are:

NeutronNumber

1

2

3

4

5

6

7

8

9

10

Number ofStable Isotopes

2

2

1

1

2

2

2

2

1

3

NeutronNumber

11

12

13

14

15

16

17

18

19

20

Number ofStable Isotopes

1

3

1

3

1

3

1

3

0

5

NeutronNumber

21

22

23

24

25

26

27

28

29

30

Number ofStable Isotopes

0

3

2

3

1

4

4

4

1

4

NeutronNumber

31

32

33

34

35

36

37

38

39

40

Number ofStable Isotopes

1

3

1

3

0

2

3

7

1

4

NeutronNumber

41

42

43

44

45

46

47

48

49

50

Number ofStable Isotopes

1

5

3

4

1

3

1

4

2

5

NeutronNumber

51

52

53

54

55

56

57

58

59

60

Number ofStable Isotopes

2

4

3

4

2

3

2

3

1

3

NeutronNumber

61

62

63

64

65

66

67

68

69

70

Number ofStable Isotopes

1

5

1

3

1

3

2

2

1

7

NeutronNumber

71

72

73

74

75

76

77

78

79

80

Number ofStable Isotopes

4

5

1

5

2

4

2

4

3

3

NeutronNumber

81

82

83

84

85

86

87

88

89

90

Number ofStable Isotopes

2

8

4

1

3

1

2

4

0

6

NeutronNumber

91

92

93

94

95

96

97

98

99

100

Number ofStable Isotopes

2

4

1

4

1

6

1

5

2

4

NeutronNumber

101

102

103

104

105

106

107

108

109

110

Number ofStable Isotopes

2

2

2

5

2

2

0

3

0

3

NeutronNumber

111

112

113

114

115

116

117

118

119

120

Number ofStable Isotopes

2

2

1

2

1

2

3

1

3

3

NeutronNumber

121

122

123

124

125

126

127

128

129

130

Number ofStable Isotopes

1

3

1

3

0

1

1

0

0

3

NeutronNumber

131

132

133

134

135

136

137

138

139

140

Number ofStable Isotopes

0

0

0

0

0

3

1

1

1

0

Note that Silicon is the
element with atomic number 14 and it has three stable isotopes whereas the elements Aluminum and Phosphorus having
atomic numbers 13 and 15, respectively, have only one stable isotope each. The isotope of silicon which is most
prevalent has atomic weight 28=14+14. The binding energy per nucleon for silicon is however not exceptionally high,
but it is the end result of a fusion process in stars called silicon burning, indicating a significant degree of
stability.

The case for the magicalness of the magic numbers is much less impressive in terms of neutron number than
for the proton numbers. In particular, there is as much of a case for 14 and 22 being magic in terms of neutron number as
for 20. Likewise the case for 8 is not notable in terms of neutron number. Fifty two seems as much
of a magic neutron number as 50. Likewise 70 would appear to be as much of a magic number as
82.

The magicality of neutron and proton numbers can be established unambiguously by looking
at the incremental binding energies of nuclides as a function of neutron number or proton
number. The incremental binding energy drops
significantly at the magic numbers. For example,

The sharp drop occurs when the neutron number is 82. The sawtooth pattern is due to the formation of neutron pairs.
For more on this topic where the signficance of the magic numbers is clearly established
and 14 is revealed to also be a magic number see